View source for Hartogs-Laurent series

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A series

$$ \tag{* }
\sum _ {k = - \infty } ^  \infty  
f _ {k} ( \prime z)
( z _ {n} - a _ {n} )  ^ {k} ,
$$

where  $  \prime z = ( z _ {1} \dots z _ {n-} 1 ) $
and where the  $  f _ {k} ( \prime z) $
are functions holomorphic in some domain  $  \prime D \subset  \mathbf C  ^ {n-} 1 $
which is independent of  $  k $. 
If  $  f _ {k} = 0 $
for all  $  k < 0 $, 
the series (*) is known as a Hartogs series. Any holomorphic function in a [[Hartogs domain|Hartogs domain]]  $  D $
of the type

$$ 
\{ {( \prime z, z _ {n} ) } : {
\prime z \in \prime D,\ 
0 \leq  r ( \prime z) < | z _ {n} - a _ {n} | <
R ( \prime z) \leq  + \infty } \}
$$

can be expanded into a Hartogs–Laurent series which converges absolutely and uniformly inside  $  D $. 
In complete Hartogs domains this will be the expansion into a Hartogs series. The domains of convergence of Hartogs–Laurent series are domains of the same kind with special  $  r ( \prime z) $
and  $  R ( \prime z) $, 
known as Hartogs radii. If  $  n = 1 $, 
when all  $  f _ {k} $
are constant, a Hartogs–Laurent series is called a [[Laurent series|Laurent series]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Behnke,   P. Thullen,   "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer  (1970)  (Elraged &amp; Revised Edition. Original: 1934)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bochner,   W.T. Martin,   "Several complex variables" , Princeton Univ. Press  (1948)</TD></TR></table>